From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8870 Path: news.gmane.org!not-for-mail From: Anders Kock Newsgroups: gmane.science.mathematics.categories Subject: Re: A construction for polynomials. Date: Wed, 6 Apr 2016 07:04:40 +0000 Message-ID: References: Reply-To: Anders Kock NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1459937543 28550 80.91.229.3 (6 Apr 2016 10:12:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 6 Apr 2016 10:12:23 +0000 (UTC) To: Nikita Danilov , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Wed Apr 06 12:12:12 2016 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.22]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ankRX-0005aS-UA for gsmc-categories@m.gmane.org; Wed, 06 Apr 2016 12:12:12 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52006) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ankRD-0002cb-Vz; Wed, 06 Apr 2016 07:11:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ankR4-00045w-4a for categories-list@mlist.mta.ca; Wed, 06 Apr 2016 07:11:42 -0300 In-Reply-To: Accept-Language: da-DK, en-US Content-Language: da-DK Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8870 Archived-At: Dear Nikita,=0A= =0A= the special significant case with which you begin, deals with the forgetful= =0A= functor (which you call P) from rings to sets. It lives in a category =0A= (topos) E of covariant functors from rings to sets, and this category you = =0A= do not give a name. But P and E seem to me to be the main actors in your = =0A= construction. What you call R[S] (for R in Rings, and S in Sets) is the =0A= value at R of the exponent object P^S -> P in E. In particular, R[1] is =0A= the value at R of the object P -> P in E. What you observe has as a =0A= special case the fact that P -> P, as an object in E, has the universal =0A= property of "P[X]", the free P-algebra in one generator; i.e.=0A= P[X] =3D (P -> P) =0A= in E. This coincidence, of a colimit type universal property (of P[X]), wit= h =0A= a limit type universal property (of the exponential P -> P), is =0A= significant, and generalizes to further duality results in E. =0A= It also generalizes to any othe algebraic theory T, not just to the theory = =0A= of rings. For instance, for T the initial algebraic theory (the algebraic = =0A= theory of sets), it implies that P+1 =3D (P -> P).=0A= For an elaboration of these generalizations, see my "Duality for Generic = =0A= Algebras", Cahiers 56 (2015), 2-14, or =0A= http://home.math.au.dk/kock/DGA03.pdf=0A= =0A= Anders=0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]